Matching of resistor sensitivities to process-induced variations in resistor widths

ABSTRACT

A method is disclosed for matching the sensitivities of different-sized resistors to changes in resistance due to changes in width resulting from a systematic manufacturing error. In order to produce sets of resistors which can be deployed in predetermined ratios of resistance, the sensitivities of a matching resistor and a reference resistor are equalized by forming the matching resistor as a plurality of parallel strips as opposed to a unitary rectangular section.

This application is a division of application Ser. No. 765,809 filedAug. 15, 1985, now U.S. Pat. No. 4,586,019, which is a division ofapplication Ser. No. 423,255, filed Sept. 24, 1982, now U.S. Pat. No.4,565,000.

BACKGROUND OF THE INVENTION

This invention relates to integrated circuit devices and, moreparticularly, to a method and a product formed thereby for matching thesensitivities of resistors to uniform manufacturing errors.

Many integrated circuit devices utilize pairs of resistors which must bedeployed in critical, predetermined ratios of resistance. The properoperation of these circuits requires that these ratios fall within verynarrow limits of error. While the ratios of resistance are critical, theindividual values of resistance of the resistors which comprise thepairs are not critical. During the formation of these resistors,however, manufacturing processes result in a uniform, systematic errorin the cutting or etching of the edges of the sections produced. Thissystematic error varies in magnitude from batch to batch of manufacturedresistors, but is uniform within any one group. Since the etching erroris always uniform within any one process group, paris of resistors canbe deployed in predetermined ratios that will remain constant even afterprocessing introduces the etching error. The design constraint ofrequiring highly accurate, predetermined ratios is thereby satisfied,although the cutting error causes the absolute values of the individualresistors in the pair to vary disproportionally.

A more general way of looking at the problem is to observe that variousmanufacturing steps affect the edges of resistors on any given chip in auniform manner. If the resistors are over-etched, for example, eachresistor will turn out to be less than the desired width by almost thesame incremental amount. Since this amount depends very little on thedesign width of the resistor, it will affect different-sized resistorsin different proportions. The resistance of wide resistors will beaffected only a little, while that of narrow ones will be affected agreat deal by the same fixed error in width.

Different-sized resistors sections, then, experience different relativechanges in resistance due to this edge error. Given two rectangularsections of equal fixed lengths and widths W and 2W, an etching error,e, which affects each pair of edges would change the resistance of thesection having width W twice as much as the resistance of the sectionhaving width 2W. The smaller of these two resistor sections is twice assensitive to changes in resistance due to changes in width. For example,given one resistor 5 mils wide and another 10 mils wide, an etchingerror which causes a variation of width of 0.03 mil on each edge wouldresult in changes in total resistance of (0.03×2)/5=1.2% and(0.03×2)/10=0.6% respectively.

Various methods can be used to circumvent this problem. For example, inthe case of R-2R networks, resistors of one size only are often employedby joining two such resistors in series to form the 2R branches of thiscircuit. Uniform edge related errors may affect the total resistance ofthis network, but will not affect the critical R-2R ratio. Other specialcases have particular solutions such as the use of numbers of identicalresistors segments to form integral ratios other than two. When large ornon-integral ratios are involved, however, fields of interconnectedidentical resistors become impractical.

An object of the present invention is to provide a method for equalizingor matching the sensitivities of different-sized resistors to relativechanges in resistance resulting from changes in width. This method isuseful in the process of manufacturing pairs of resistors which musthave predetermined ratios of resistance within strict error limits. Thisratio of resistance must be held at a predetermined constant despite themanufacturing error described above which introduces variations in widthalong each edge of the resistors and also changes the resistance of theresistors. Changes in the resistance of each resistor in the pair areacceptable so long as the ratio of the two resistance values remainsconstant after manufacture is completed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a pictorial showing of one embodiment of the invention; and

FIG. 2 is a pictorial showing of a modified form of the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENT

The method and product formed thereby which are disclosed may beutilized to match the sensitivities of different-sized resistors. Forany rectangular section of resistor material, the resistance is given bythe expression:

    R=(ρL)/W.                                              Equation 1

where

p=resistivity of the sheet

L=length of the section, and

W=width of the section.

The change in resistance due to a change in width is dR/dW, the firstderivative of R with respect to W. For this specification and theappended claims, the term "sensitivity," denoted as S, is the quantitywhich serves as an index of the degree of variation of the totalresistance of a resistor due to changes in its width. The sensitivity ofa resistor, S, is defined as: ##EQU1##

For a resistor with i rectangular sections each of the same lengthhaving combined width

    W.sub.1 +W.sub.2 +W.sub.3 + . . . +W.sub.i =W,

the total sensitivity is given by the sum of the sensitivities of eachsection: ##EQU2##

For the more complex case of a resistor with i rectangular sections inseries, each of different length and different width, the totalsensistivity is given by another form of Equation 2:

Since for any section ##EQU3##

Which, by substitution, yields ##EQU4##

Throughout this specification, the method of the invention will beexplained using a pair of resistors called a reference resistor, R_(r),and a matching resistor, R_(m). Given a pair of resistors R_(r) andR_(m) each having length L, resistivity ρ, widths a and b respectively,the sensitivities of R_(r) and R_(m) would be, according to Equation 2,

    S.sub.r =-1/a;

    S.sub.m =-1/b.

In order to match the sensitivities, the ratio to S_(r) to S_(m) must bemade to equal one. This is accomplished by dividing the larger of thetwo resistors R_(r) or R_(m) into n parallel sections where ##EQU5##

If a wide resistor having low sensitivity to edge effects is to bematched to a narrower resistor with higher sensitivity, the wideresistor can be divided to increase its sensitivity. For example, giventwo resistors R_(x) and R_(y) of equal lengths L and widths of 4.0 milsand 0.5 mil respectively, Equation 2 yields

    S.sub.x =-1/4.0 mil=-0.25 mil.sup.-1,

    S.sub.y =-1/0.5 mil=-2.0 mil.sup.-1.

In order to match these two resistors, the wider resistor R_(x) would bedivided into eight parallel sections, as shown at 10 in FIG. 1. Thisresults from the following calculation:

From Equation 5, ##EQU6## These 8 sections may each be of equal width,i.e. 0.5 mil, or may be of different widths, depending upon layoutrequirements. The matching process, however, requires that the divisionin this case result in 8 sections. That is, the resistor which resultsfrom the division will have 8 times as many edges. Resistors R_(x),which had 2 edges, will have 16 edges after division.

A more complex problem is presented by a resistor pair consisting of afirst resistor having two series sections of different lengths andwidths and a second having one section. If the first resistor, R_(c),consists of sections 17.8 mils long by 0.5 mil wide and 9 mils long by1.4 mils wide, the resistors will have separate resistance values of35.60kΩ and 6.43KΩ respectively for a given, common resistivity. Due toan added discontinuity resistance of 0.42kΩ attributable to the junctionof the two resistors, the combined series resistance of the two sectionsis 35.062k106 +6.43kΩ+0.42kΩ=42.45kΩ. From Equation 4, ##EQU7## Ifresistor R_(c) is to match a 250Ω resistor, R_(d), which is 4 mils long,the 250Ω resistor must be divided. The resistor is 16 mils wide so thatthe sensitivity is 1/16=0.0625. The sensitivity of this resistor is1.79/0.0625=28.64 times less than the 42kΩ resistor. Therefore, the 250Ωresistor should be divided into 28 or 29 sections.

Although some inaccuracy is introduced in the example above by allowingthe selection of either 28 or 29 divisions, the error is relativelyslight. If, however, the number of divisions, n, is a relatively lownumber, such as 2.64, the error introduced by selecting n=2 or n=3 couldbe unacceptably high. To avoid that result, one of the parallel resistorsections could be formed of two series-arranged sub-sections ofdifferent widths. For example, as shown in FIG. 2, the section 12 couldcomprise adjoining rectangles 14 and 16 having parallel edges butdifferent widths. Such a configuration would enable the designer tofabricate one section of non-uniform width which could contributefractional sensitivity: one value of sensitivity due to its narrow endand another, different value of sensitivity due to its wide end. Thisarrangement could be utilized to account for the non-integral portion ofthe sensitivity ration, n, when that ratio is relatively low.

It is believed that the many advantages of this invention will now beapparent to those skilled in the art. It will also be apparent that anumber of variations and modifications may be made in this inventionwithout departing from its spirit and scope. Accordingly, the foregoingdescription is to be construed as illustrative only, rather thanlimiting. This invention is limited only by the scope of the followingclaims.

I claim:
 1. In the art of making integrated-circuit chips havingresistors thereon, a method of controlling the resistance ratio betweena first resistor and a second resistor on the integrated-circuit chip,wherein said first resistor if formed as a single element would have asensitivity of resistance change due to changes in width which issubstantially less than the corresponding sensitivity of said secondresistor, said resistance ratio being controlled by matching thesensitivity of said first resistor to changes in resistance due tochanges in width to the corresponding sensitivity of said secondresistor by a process wherein said first resistor is divided into aplurality of parallel-connected sections having a total width providingthe same resistance as said single element but presenting an increasednumber of side edges relative to said single element to provide that thecomposite sensitivity of said parallel-connected resistor sections isincreased and made at least substantially equal to said sensitivity ofsaid second resistor;the improved method comprising the steps of:calculating the ratio of (1) said sensitivity of said second resistor to(2) said sensitivity of said hypothetical single-element first resistor;and setting the number of side edges to be presented by saidparallel-connected sections to be at least substantially equal to twicesaid ratio.